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Since the 60 GHz channel has been shown to be frequency selective for very large bandwidths and low antenna gains [3,4], or-thogonal frequency division multiplexing OFDM has been propose

EURASIP Journal on Wireless Communications and Networking

Volume 2007, Article ID 86206, 14 pages

doi:10.1155/2007/86206

Research Article

Comparison of OQPSK and CPM for Communications at

60 GHz with a Nonideal Front End

Jimmy Nsenga, 1, 2 Wim Van Thillo, 1, 2 Franc¸ois Horlin, 1 Andr ´e Bourdoux, 1 and Rudy Lauwereins 1, 2

1 IMEC, Kapeldreef 75, 3001 Leuven, Belgium

2 Departement Elektrotechniek - ESAT, Katholieke Universiteit Leuven, Kasteelpark Arenberg 10, 3001 Leuven, Belgium

Received 4 May 2006; Revised 14 November 2006; Accepted 3 January 2007

Recommended by Su-Khiong Yong

Short-range digital communications at 60 GHz have recently received a lot of interest because of the huge bandwidth available

at those frequencies The capacity offered to the users could finally reach 2 Gbps, enabling the deployment of new multimedia applications However, the design of analog components is critical, leading to a possible high nonideality of the front end (FE) The goal of this paper is to compare the suitability of two different air interfaces characterized by a low peak-to-average power ratio (PAPR) to support communications at 60 GHz On one hand, we study the offset-QPSK (OQPSK) modulation combined with a channel frequency-domain equalization (FDE) On the other hand, we study the class of continuous phase modulations (CPM) combined with a channel time-domain equalizer (TDE) We evaluate their performance in terms of bit error rate (BER) considering a typical indoor propagation environment at 60 GHz For both air interfaces, we analyze the degradation caused by the phase noise (PN) coming from the local oscillators; and by the clipping and quantization errors caused by the analog-to-digital converter (ADC); and finally by the nonlinearity in the PA

Copyright © 2007 Jimmy Nsenga et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

We are witnessing an explosive growth in the demand for

wireless connectivity Short-range wireless links like

wire-less local area networks (WLANs) and wirewire-less personal

area networks (WPANs) will soon be expected to deliver

bit rates of over 1 Gbps to keep on satisfying this demand

Fast wireless download of multimedia content and

stream-ing high-definition TV are two obvious examples As lower

frequencies (below 10 GHz) are getting completely congested

though, bandwidth for these Gbps links has to be sought at

higher frequencies Recent regulation assigned a 3 GHz wide,

worldwide available frequency band at 60 GHz to this kind of

applications [1]

Communications at 60 GHz have some advantages as

well as some disadvantages The main advantages are

three-fold The large unlicensed bandwidth around 60 GHz (more

than 3 GHz wide) will enable very high data rate wireless

ap-plications Secondly, the high free space path loss and high

attenuation by walls simplify the frequency reuse over small

distances Thirdly, as the wavelength in free space is only

5 mm, the analog components can be made small Therefore,

on a small area, one can design an array of antennas, which

steers the beam in a given target direction This improves the link budget and reduces the time dispersion of the channel Opposed to this are some disadvantages: the high path loss will restrict communications at 60 GHz to short distances, more stringent requirements are put on the analog com-ponents (like multi-Gsamples/s analog-to-digital converter ADC), and nonidealities of the radio frequency (RF) front end have a much larger impact than at lower frequencies The design of circuits at millimeter waves is more problematic than at lower frequencies for two important reasons First, the operating frequency is relatively close to the cut-off fre-quency and to the maximum oscillation frefre-quency of nowa-days’ complementary metal oxide semiconductor (CMOS) transistors (e.g., the cut-off frequency of a transistor in a

90 nm state-of-the-art CMOS is around 150 GHz [2]), reduc-ing significantly the design freedom Second, the wavelength approaches the size of on-chip dimensions so that the inter-connects have to be modeled as (lossy) transmission lines, complicating the modeling and circuit simulation and also the layout of the chip

A suitable air interface for low-cost, low-power 60 GHz transceivers should thus use a modulation technique that has

a high level of immunity to FE nonidealities (especially phase

noise (PN) and ADC quantization and clipping), and allows

an efficient operation of the power amplifier (PA) Since the

60 GHz channel has been shown to be frequency selective

for very large bandwidths and low antenna gains [3,4],

or-thogonal frequency division multiplexing (OFDM) has been

proposed for communications at 60 GHz However, it is very

sensitive to nonidealities such as PN and carrier frequency

offset (CFO) Moreover, due to its high PAPR, it requires the

PA to be backed off by several dB more than for a single

car-rier (SC) system, thus lowering the power efficiency of the

system

Therefore, we consider two other promising air interfaces

that relax the FE requirements First, we study an SC

transmission scheme combined with OQPSK because it

has a lower PAPR than regular QPSK or QAM in

band-limited channels As the multipath channel should be

equalized at a low complexity, we add redundancy at the

transmitter to make the signal cyclic and to be able to

equalize the channel in the frequency domain [5]

Sec-ondly, we study CPM techniques [6] These have a

per-fectly constant amplitude, or a PAPR of 0 dB Moreover,

their continuous phase property results in lower spectral

sidelobes Linear representations and approximations

de-veloped by Laurent [7] and Rimoldi [8] allow for great

complexity reductions in the equalization and detection

processes In order to mitigate the multipath channel,

a conventional convolutive zero-forcing (ZF) equalizer is

used

The goal of this paper is to analyze, by means of

simula-tions, the impact of three of the most critical building blocks

in RF transceivers, and to compare the robustness of the two

air interfaces to their nonideal behavior:

(i) the mixing stage where the local oscillator PN can be

very high at 60 GHz,

(ii) the ADC that, for low-power consumption, must have

the lowest possible resolution (number of bits) given

the very high bit rate,

(iii) the PA where nonlinearities cause distortion and

spec-tral regrowth

The paper is organized as follows InSection 2, we

de-scribe the indoor channel at 60 GHz Section 3 describes

the considered FE nonidealities Sections 4 and 5

intro-duce the OQPSK and CPM air interfaces, respectively,

to-gether with their receiver design Simulation setup and

re-sults are provided inSection 6and the conclusions are drawn

inSection 7

Notation

We use roman letters to represent scalars, single underlined

letters to denote column vectors, and double underlined

let-ters to represent matrices [·]T and [·]H stand for

trans-pose and complex conjugate transtrans-pose operators,

respec-tively The symbol denotes the convolution operation and

⊗the Kronecker product I k is the identity matrix of size

k × k and 0 m × n is anm × n matrix with all entries equal

to 0

2 THE INDOOR 60 GHZ CHANNEL

2.1 Propagation characteristics

The interest in the 60 GHz band is motivated by the large amount of unlicensed bandwidth located between 57 and

64 GHz [1, 9] Analyzing the spectrum allocation in the United States (US), Japan, and Europe, one notices that there

is a common contiguous 3 GHz bandwidth between 59 and

62 GHz that has been reserved for high data rate applications This large amount of bandwidth can be exploited to establish

a wireless connection at more than 1 Gbps

Different measurement campaigns have been carried out

to characterize the 60 GHz channel The free space loss (FSL) can be computed using the Friis formula (1) as follows:

FSL [dB]=20×log10



4πd λ



whereλ is signal wavelength and d is the distance of the

ter-minal from the transmitter base station One can see that the FLS is already 68 dB at 1 m separation away from the trans-mitter Thus, given the limited transmitted power, the com-munication range will hardly extend over 10 m Besides the FSL, reflection and penetration losses of objects at 60 GHz are higher than at lower frequencies [10,11] For instance, concrete walls 15 cm thick attenuate the signal by 36 dB They act thus as real boundaries between different rooms

However, the signals reflected off the concrete walls have

a sufficient amplitude to contribute to the total received power, thus making the 60 GHz channel a multipath chan-nel [3,12] Typical root mean-square (RMS) delay spreads at

60 GHz can vary from 10 nanoseconds to 100 nanoseconds

if omni-directional antennas are used, depending on the di-mensions and reflectivity of the environment [3] However, the RMS delay spread can be greatly reduced to less than 1 nanosecond by using directional antennas, thus increasing the coherence bandwidth of the channel up to 200 MHz [13] Moreover, the objects moving within the communica-tion environment make the channel variant over time Typ-ical values of Doppler spread at 60 GHz are around 200 Hz

at a normal walking speed of 1 millisecond This results in

a coherence time of approximatively 1 millisecond With a symbol period of 1 nanosecond, 106 symbols can be trans-mitted in a quasistatic environment Thus, Doppler spread

at 60 GHz will not have a significant impact on the system performance

In summary, 60 GHz communications are mainly suit-able for short-range communications due to the high prop-agation loss The channel is frequency selective due to the large bandwidth used (more than 1 GHz) However, one can assume the channel to be time invariant during the transmis-sion of one block

2.2 Channel model

In this study, we model the indoor channel at 60 GHz us-ing the Saleh-Valenzuela model [14], which assumes that the

received signals arrive in clusters The rays within a cluster

have independent uniform phases They also have

indepen-dent Rayleigh amplitudes whose variances decay

exponen-tially with cluster and rays delays In the Saleh-Valenzuela

model, the cluster decay factor is denoted byΓ and the rays

decay factor is represented by γ The clusters and the rays

form Poisson arrival processes that have different, but fixed

ratesΛ and λ, respectively [14]

We consider the same scenario as that defined in [15]

The base station has an omni-directional antenna with 120◦

beam width and is located in the center of the room The

re-mote station has an omni-directional antenna with 60◦beam

width and is placed at the edge of the room The

correspond-ing Saleh-Valenzuela parameters are presented inTable 1

3 NONIDEALITIES IN ANALOG TRANSCEIVERS

In this section, we introduce 3 FE nonidealities: ADC

clip-ping and quantization, PN and nonlinearity of the PA The

rationale for choosing these 3 nonidealities is that a good PA,

a high resolution ADC, and a low PN oscillator have a high

power consumption [16]

3.1 Clipping and quantization

3.1.1 Motivation

The number of bits (NOB) of the ADC must be kept as low

as possible for obvious reasons of cost and power

consump-tion On the other hand, a large number of bits is desirable

to reduce the effect of quantization noise and the risk of

clip-ping the signal Clipclip-ping occurs when the signal fluctuation

is larger than the dynamic range of the ADC Without going

into detail, we mention that there is always an optimal

clip-ping level for a given NOB As the clipclip-ping level is increased,

the signal degradation due to clipping is reduced However,

the degradation due to quantization is increased as a larger

dynamic range must be covered with the same NOB For a

more elaborate discussion, we refer to [17]

3.1.2 Model

The ADC is thus characterized by two parameters: the NOB

and the normalized clipping levelμ, which is the ratio of the

clipping level to the RMS value of the amplitude of the signal

InFigure 1, we illustrate the clipping/quantization function

for an NOB =3 This simple model is used in our

simula-tions inSection 6.4

3.2 Phase noise

3.2.1 Motivation

PN originates from nonideal clock oscillators,

voltagecon-trolled oscillators (VCO), and frequency synthesis circuits

In the frequency domain, PN is most often characterized by

the power spectral density (PSD) of the the oscillator phase

φ(t) The PSD of an ideal oscillator has only a Dirac pulse at

its carrier frequency, corresponding to no phase fluctuation

Table 1: Saleh-Valenzuela channel parameters at 60 GHz

1/Λ 75 nanoseconds

Γ 20 nanoseconds 1/λ 5 nanoseconds

γ 9 nanoseconds

NormalizedVout

NormalizedVin



Vin RMS(Vin )



Figure 1: ADC input-output characteristic

at all In practice, the PSD of the phase exhibits a 20 dB/dec decreasing behavior as the offset from the carrier frequency increases Nonmonotonic behavior is attributable to, for ex-ample, phase-locked loop (PLL) filters in the frequency syn-thesis circuit

3.2.2 Model

We characterize the phase noise by a set of 3 parameters (see Figure 2) [18]:

(i) the integrated PSD denotedK, expressed in dBc, which

is the two-sided integral of the phase noise PSD, (ii) the 3 dB bandwidth,

(iii) the VCO noise floor

Note that these 3 parameters will fix the value of the PN PSD

at low frequency offsets In our simulations (seeSection 6.3),

we assume a phase noise bandwidth of 1 MHz and a noise floor of−130 dBc/Hz Typical values of the level of PN PSD

at 1 MHz are considered [19] and the corresponding inte-grated PSD is calculated in Table 2 In order to generate a phase noise characterized by the PSD illustrated inFigure 2,

a white Gaussian noise is convolved with a filter whose fre-quency domain response is equal to the square root of the PSD

3.3 Nonlinear power amplification

3.3.1 Motivation

Nonlinear behavior can occur in any amplifier but it is more likely to occur in the last amplifier of the transmitter where the signal power is the highest For power consumption rea-sons, this amplifier must have a saturated output power that

is as low as possible, compatible with the system level con-straints such as transmit power and link budget The gain characteristic of an amplifier is almost perfectly linear at low

Table 2: Simulated integrated PSD.

PN @1 MHz [dBc/Hz] Integrated PSD [dBc]

O ffset from carrier

3 dB cut-o ff

−20 dB/decade

Noise floor

Figure 2: Piecewise linear phase noise PSD definition used in the

phase noise model

input level and, for increasing input power, deviates from

the linear behavior as the input power approaches the 1-dB

compression point (P1 dB: the point at which the gain is

re-duced by 1-dB because the amplifier is driven into

satura-tion) and eventually reaches complete saturation The input

third-order intercept point (IP3) is also often used to

quan-tify the nonlinear behavior of amplifiers It is the input power

at which the power of the two-tone third-order

intermodu-lation product would become equal to the power of the

first-order term When peaks are present in the transmitted

wave-form, one has to operate the PA with a few dBs of backoff to

prevent distortion This backoff actually reduces the power

efficiency of the PA and must be kept to a minimum

3.3.2 Model

In our simulation (seeSection 6.5), we characterize the

non-linearity of the PA by a third-order nonlinear equation

y(t) = a1x(t) + a3x(t)2

where x(t) and y(t) are the baseband equivalent PA input

and output, respectively,a1anda3are real polynomial

coef-ficients We assume an amplifier with a unity gain (a1 =1)

and an input amplitude at 1-dB compression pointA1dB

nor-malized to 1 Therefore, by using (3), one can compute the

third-order coefficient a3

a3= −0.145 a1

A2

1 dB

Ain 1

xRMS Backoff > 0

yRMS 1

Aout

1 dB

Figure 3: PA input-output power characteristic

The parametera3is then equal to−0.145 Note that (2) models only the amplitude-to-amplitude (AM-AM) conver-sion of a nonlinear PA In order to make our model more realistic, a saturation level is set from the extremum of the cubic function The root mean-square (RMS) value of the in-put PA signal is comin-puted and its level is adapted according

to the backoff requirement The backoff is defined relative to

A1 dBand is the only varying parameter Then the nonlinear-ity is introduced using the AM-AM conversion as shown in Figure 3

4 OFFSET QPSK WITH FREQUENCY DOMAIN EQUALIZATION

4.1 Initial concept

Offset-QPSK, a variant of QPSK digital modulation, is char-acterized by a half symbol period delay between the data mapped on the quadrature (Q) branch and the one mapped

on the inphase (I) branch This offset imposes that either the

I or the Q signal changes during the half symbol period

Con-sequently, the phase shift between two consecutive OQPSK symbols is limited to ±90◦ (±180◦ in conventional QPSK modulation), thus avoiding the amplitude of the signal to go through the “0” point The advantage of an OQPSK mod-ulated signal over QPSK signal is observed in band-limited channels where nonrectangular pulse shaping, for instance, root raised root cosine, is used The envelope fluctuation of

an OQPSK signal is found to be 70% lower than that of a conventional QPSK signal [20] Thus, OQPSK is considered

to be a low PAPR modulation scheme, for which a nonlinear

PA with less backoff can be used, thus increasing the power efficiency of the system

4.2 System model

Our system model is inspired from the model of Wang and Giannakis [21] Let us consider the baseband block trans-mitter model represented in Figure 4 The inphase compo-nent of the digital OQPSK signal is denoted by u I[ k] and

u I[k]

u Q[k]

S/P

S/P

u I[k]

u Q[k]

T cp

T cp x I[k]

x Q[k]

P/S

P/S

x I[k]

x Q[k]

g Q T(t)

g T

I(t)

s(t)

Figure 4: Offset QPSK block transmission

the quadrature-phase component denoted byu Q[ k] The two

streams are first serial-to-parallel (S/P) converted to form

blocks u I[k] : = [u I[ kB], u I[ kB + 1], , u I[ kB + B −1]]T

and u Q[k] : = [u Q[kB], u Q[ kB + 1], , u Q[ kB + B −1]]T

whereB is the block size Then, a cyclic prefix (CP) of length

Ncpis inserted at the beginning of each block to get cyclic

blocksx I[k] and x Q[k] The cyclic prefix insertion is done

by multiplying bothu I[k] and u Q[k] with the matrix Tcp =

[0N

cp×(B − Ncp ),I N

cp;I B] of size (B + Ncp)× B In a practical

sys-tem, theNcp should be larger than the channel impulse

re-sponse length, and the size of the blockB is chosen so that

the CP overhead is limited (practically an overhead of 1/5 is

often used) The sizeB should on the other hand be as small

as possible to reduce the complexity and to ensure that the

channel is constant within one symbol block duration The

cyclic blocksx I[k] and x Q[k] are afterwards converted back

to serial streams and the resulting streamsx I[ k] and x Q[ k] of

sample duration equal toT are filtered by square root raised

cosine filtersg I T(t) and g Q T(t), respectively The inherent offset

betweenI and Q branches, which differentiates the OQPSK

signaling from the normal QPSK, is modeled through the

pulse-shaping filters defined such thatg T

I(t − T/2).

The two pulse-shaped signals are then summed together to

form the equivalent complex lowpass transmitted signals(t).

The signals(t) is then transmitted through a frequency

selective channel, which we model by its equivalent lowpass

channel impulse responsec(t).Figure 5shows a block

dia-gram of the receiver The received signalrin(t) is corrupted

by additive white Gaussian noise (AWGN),n(t), generated

by analog FE components The noisy received signal is first

lowpass-filtered by an anti-aliasing filter with ideal lowpass

specifications before the discretization We consider the

fol-lowing two sample rates

(i) The nonfractional sampling (NFS) rate which

corre-sponds to sampling the analog signal everyT seconds.

The corresponding anti-aliasing filter, denotedg R

NFS(t),

eliminates all the frequencies above 0.5/T.

(ii) The fractional sampling (FS) rate for which the

sam-pling period is T/2 seconds The cutoff frequency of

the anti-aliasing filterg R

FS(t) is set to 1/T.

More information about the two sampling modes can be

found in [22] In the sequel, we focus on the FS case The

NFS can be seen as a special case of FS In order to

character-ize the received signal, we defineh I( t) : = g T

I(t) c(t)g R

FS(t)

andh Q( t) : = j ∗ g Q T(t)  c(t)  g R

FS(t) as the overall

chan-nel impulse response encountered by data symbols onI and

Q, respectively The received signal after lowpass filtering is

given by

r(t) =

k

x I[ k]h I( t − kT) +

k

x Q[k]h Q( t − kT) + v(t)

(4)

in which v(t) is the lowpass filtered noise The analog

re-ceived signalr(t) is then sampled every T/2 seconds to get

the discrete-time sequencer[m].

As explained in [22], fractionally sampled signals are pro-cessed by creating polyphase components, where even and odd indexed samples of the received signal are separated In the following, the index “0” is related to even samples or polyphase component “0” while odd samples are represented

by index “1” or polyphase component “1.” Thus, we define

r ρ[m]def= r[2m + ρ],

v ρ[m]def= v[2m + ρ],

h ρ I[m]def= h I[2 m + ρ],

h ρ Q[m]def= h Q[2 m + ρ],

(5)

whereρ denotes either the polyphase component “0” or the

polyphase component “1,”r[m] and v[m] are, respectively,

the received signalr(t) and the noise v(t) sampled every T/2

seconds, h I[ m] and h Q[ m] represent the discrete-time

ver-sion of, respectively,h I( t) and h Q(t) sampled every T/2

sec-onds The sampled channelsh ρ I[m] and h ρ Q[m] have finite

impulse responses, of lengthL I andL Q, respectively These

time dispersions cause the intersymbol interference (ISI) be-tween consecutive symbols, which, if not mitigated, degrades the performance of the system Next to the separation in polyphase components, we separate the real and imaginary parts of different polyphase signals Starting from now, we use the supplementary upper indexc = { r, i }to identify the real or imaginary parts of the sequences

The four real-valued sequences r ρc[m] are serial

to parallel converted to obtain the blocks r ρc[m] :=

[r ρc[mB], r ρc[mB+1], , r ρc[mB+B+Ncp−1]]Tof (B+Ncp) samples The corresponding transmit-receive block relation-ship, assuming a correct time and frequency synchroniza-tion, is given by

r ρc[m] = H ρc I [0]Tcpu I[m] + H ρc I [1]Tcpu I[m −1]

+H ρc Q[0]Tcpu Q[m] + H ρc Q[1]Tcpu Q[m −1] +v ρc[m],

(6)

where v ρc[m] is the mth filtered noise block defined as

v ρc[m] : =[v ρc[mB], v ρc[mB +1], , v ρc[mB +B +Ncp−1]]T The square matricesH ρc X[0] andH ρc X[1] of size (B+Ncp)×(B+

N ), withX equal to I or Q, are represented in the following

rin (t)

n(t)

gNFSR (t)

T r(t) r[m]

Real Imag.

S/P

S/P

r r[m]

r i[m]

Rcp

Rcp

y r[m]

y i[m]

rin (t)

n(t)

gFSR(t) T/2 r(t) r[m]

z −1

2

2

Real

r0 [m]

Imag.

Real

r1 [m]

Imag.

S/P

S/P S/P

S/P

r0r[m]

r0i[m]

r1r[m]

r1i[m]

Rcp

Rcp

Rcp

Rcp

y0r[m]

y0i[m]

y1r[m]

y1i[m]

Figure 5: Receiver: upper part NFS, lower part FS

equations:

H ρc X[0]=

⎡

⎢

⎢

⎢

⎢

⎢

h ρc X[0] 0 0 · · · 0

h ρc X[0] 0 · · · 0

h ρc X[L X] · · · · · · 0

· · · 0

0 · · · h ρc X[L X] · · · h ρc X[0]

⎤

⎥

⎥

⎥

⎥

⎥

,

H ρc

⎡

⎢

⎢

⎢

⎢

⎢

0 · · · h ρc X[L X] · · · h ρc X[1]

0 · · · · · · h ρc

. · · · .

0 · · · 0 · · · 0

⎤

⎥

⎥

⎥

⎥

⎥

.

(7)

The second and the fourth terms in (6) highlight the

inter-block interference (IBI) that arises between consecutive

blocks due to the time dispersion of the channel The IBI

between consecutive blocks u I[m] or u Q[m] is afterwards

eliminated by discarding the first Ncp samples in each

re-ceived block This operation is carried out by multiplying

the received blocks in (6) by a guard removal matrixRcp =

[0

B × Ncp,I B] of sizeB ×(B + Ncp) We get

y ρc[m]def= Rcpr ρc[m] = RcpH ρc I [0]Tcpu I[m]

+RcpH ρc

+RcpH ρc Q[1]Tcpu Q[m −1] +Rcpv ρc[m].

(8)

AsNcphas been chosen to be larger than the max{ L I, L Q },

the product of Rcp and each of H ρc I [1] andH ρc Q[1]

matri-ces is null Moreover, the left and right cyclic prefix

inser-tion and removal operainser-tions aroundH ρc I [0] andH ρc Q[0],

de-scribed mathematically asR H ρc[0]T andR H ρc[0]T ,

respectively, result in circulant matrices ˙H ρc I and ˙H ρc Q of size (B × B) Finally, the discrete-time block input-output

rela-tionship taking the CP insertion and removal operations into account is

y ρc[m] = H˙ρc I u I[m] + ˙ H ρc Q u Q[m] + w ρc[m] (9)

in whichw ρc[m] is obtained by discarding the first Ncp sam-ples from the filtered noise blockv ρc[m] By stacking the real

and the imaginary parts of the two polyphase components

on top of each other, the matrix representation of the FS case is

⎡

⎢

⎢

y0r[m]

y0i[m]

y1r[m]

y1i[m]

⎤

⎥

⎥

  

y[m]

=

⎡

⎢

⎢

⎢

⎣

˙

H0I r H˙0Q r

˙

H0I i H˙0Q i

˙

H1I r H˙1Q r

˙

H1I i H˙1Q i

⎤

⎥

⎥

⎥

⎦

  

˙

H



u I[m]

u Q[m]



  

u[m]

+

⎡

⎢

⎢

w0r[m]

w0i[m]

w1r[m]

w1i[m]

⎤

⎥

⎥

  

w[m]

Finally, we get

in which y[m] denotes the compound received signal, u[m]

is a vector containing both theI and Q transmitted symbols,

andw[m] denotes the noise vector, ˙ H is the compound

chan-nel matrix The vectorsy[m] and w[m] contain 4B symbols,

˙

H is a matrix of size 4B ×2B, and u[m] is a vector of 2B

symbols Notice that all these vectors and matrices are real valued Interestingly, the NFS case can be obtained from the

FS by the two following adaptations

(i) First, one has to change the analog anti-aliasing filter

at the receiver In fact, the cut-off frequency of the NFS filter is 0.5/T while it is 1/T for the FS filter.

(ii) Second, one keeps only the polyphase component with superscript index “0” in (10)

2B

B

B

1 0 0 0 · · ·0 0

0 0 1 0 · · ·0 0

0 0 0 0 · · ·0 0

0 0 0 0 · · ·1 0

0 1 0 0 · · ·0 0

0 0 0 1 · · ·0 0

0 0 0 0 · · ·0 0

0 0 0 0 · · ·0 1

P H

2 =

4B

1 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0

0 1 0 0 0 0 0 0

0 0 0 0 0 1 0 0

.

0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 1 Figure 6: Permutation matricesP and P H

At this point, even as the IBI has been eliminated between

consecutive blocks, ISI within each individual block is still

present However, the IBI-free property of the resulting

blocks allow to equalize each block independently from the

others In the following, we design an FDE to mitigate the

remaining ISI

4.3 Frequency domain equalization

According to [23], the expression of a linear minimum

mean-square error (MMSE) detector that multiplies the

re-ceived signaly[m] to provide the estimation u[m] of the vec-

tor of transmitted symbols is given by

ZMMSE=



σ2

w

σ2

u

I2B+ ˙H H H˙

−1

˙

whereσ2

wrepresent the variances of the real and

imag-inary parts of the transmitted symbols and of the AWGN,

respectively However, the computation of this expression

is very complex due to the structure of ˙H Fortunately, by

exploiting the properties of the circulant matrices

compos-ing ˙H, the latter can be transformed in a matrixΛ (of the

same size as ˙H) of diagonal submatrices, by the discrete block

Fourier transform matricesFmandFH

n defined as

˙

Fmdef

= F ⊗ I m,

FH n

def

= F H ⊗ I n,

(14)

whereF is the discrete Fourier transform matrix of size B × B.

For the FS case,m =2 andn =4 while in the NFS casem =

n =2 Note thatF mandF nare square matrices of sizemB ×

mB and nB × nB, respectively The different diagonal matrices

are denoted byΛρc

X, and their diagonals are calculated by diag

Λρc X



= √1

withh ρc X =[h ρc X[0],h ρc X[1], , h ρc X[L X]] T

In addition to the frequency domain transformation, a

permutation between columns and lines ofΛ is performed

to simplify the complexity of the matrix inversion operation

The permutation is realized such thatΛ is transformed into

Ψ=

⎡

⎢

⎢

⎢

⎢

⎢

⎢

Ψ2 .

Ψl

ΨB

⎤

⎥

⎥

⎥

⎥

⎥

⎥

withΨl =

⎡

⎢

⎢

⎢

λ0r I,l λ0r Q,l

λ0i I,l λ0i Q,l

λ1r I,l λ1r Q,l

λ1i I,l λ1i Q,l

⎤

⎥

⎥

⎥

Figure 7: Block diagonal matrix

a block diagonal matrixΨ (seeFigure 7) Thelth block Ψl contains thelth subcarrier frequency response λ ρc X,lof the dif-ferent channels; thus each subcarrier is equalized individually and independently from the others We obtain

where the permutation matricesP1 andP H

2 are defined as shown inFigure 6

Finally, by replacing (16) and (13) in (12), the expression

of the joint MMSE detector becomes

ZMMSE=FH



σ2

w

σ2

u

I +ΨHΨ−1ΨH P1Fm (17)

From (17), one derives the expression of the joint zero forc-ing (ZF) detector by assumforc-ing a very high signal-to-noise ra-tio (SNR), whereby the termσ2

ubecomes negligible [23]:

ZZF=FH

2[ΨHΨ]−1ΨH P1Fm (18) The complexity in terms of number of operations (NOPS)

of the FD equalizer computation and the equalization is as-sessed in Tables3and4, respectively The complexity of an FFT of size B is proportional to (B/2)log2B The NFS case

is much less complex than the FS It is well known that the complexity of the FDE is much smaller than the complexity

of the TDE (the inversion of the inner matrix necessary to compute the equalizer and the multiplication of the received vector by this equalizer would be both proportional toB3)

Table 3: Equalizer computation.

Task Operation NOPS

FS NFS Computation of diag(Λρc

Computation of [ΨHΨ]−1ΨH + and× 8B 4B

Table 4: Equalization

FS NFS Frequency components ofy ρc[m] FFT 4 2

Equalized symbols in time domain IFFT 2 2

5.1 Transmitted signal

CPM covers a large class of modulation schemes with a

con-stant amplitude, defined by

s(t, a) =



2E S

where s(t, a) is the sent complex baseband signal, E S the

energy per symbol, T the symbol duration, and a =

[a[0], a[1], , a[N − 1]] is a vector of length N

con-taining the sequence of M-ary data symbols a[n] =

±1,±3, , ±(M −1) The transmitted information is

con-tained in the phase

φ(t, a) =2πh

whereh is the modulation index and

q(t) =

t

Normally the functiong(t) is a smooth pulse shape over

a finite time interval 0 ≤ t ≤ LT and zero outside Thus

L is the length of the pulse per unit T The function g(t) is

normalized such that∞

−∞ g(t)dt =1/2 This means that for

schemes with positive pulses of finite length, the maximum

phase change over any symbol interval is (M −1)hπ.

As shown in [24], the BER can be halved by precoding

the information sequence before passing it through the CPM

modulator Ifd =[d[1], d[2], , d[N −1]] is a vector

con-taining the uncoded input bipolar symbol stream, the output

of the precodera (assuming M =2) can be written as

whered[ −1]=1

A conceptual general transmitter structure based on (19)

and (22) is shown inFigure 8

d[n]

Precoder a[n]

g(t) filter

2πh

FM-modulator s(t, a)

Figure 8: Conceptual modulator for CPM

5.2 GMSK for low-cost, low-power 60 GHz transmitters

GMSK has been adopted as the modulation scheme for the European GSM system and for Bluetooth due to its spectral

efficiency and constant-envelope property [25] These two characteristics result in superior performance in the pres-ence of adjacent channel interferpres-ence and nonlinear ampli-fiers [24], making it a very attractive scheme for 60 GHz ap-plications too GMSK is obtained by choosing a Gaussian fre-quency pulse

g(t) = Q



2πB T √(t − T/2)

ln 2



− Q



2πB T √(t + T/2)

ln 2



, (23)

whereQ(x) is the well-known error function and B T is the

bandwidth parameter, which represents the −3 dB bandwidth

of the Gaussian pulse We will focus on a GMSK scheme

with time-bandwidth product B T T = 0.3, which enables us

to truncate the Gaussian pulse toL =3 without significantly influencing the spectral properties [26] A modulation index

h = 1/2 is chosen as this enables the use of simple

MSK-type receivers [27] The number of symbol levels is chosen as

M =2

5.3 Linear representation by Laurent

Laurent [7] showed that a binary partial-response CPM sig-nal can be represented as a linear combination of 2L −1 ampli-tude modulated pulsesC k( t) (with t = NT + τ, 0 ≤ τ < T):

s(t, a) =

2L−1−1

where

C k( t − nT) = S(t) ·

S

t + (n + Lβ n,k) T

,

α k[ n] =

n



a[m] −

a[n − m]β n,k,

(25)

andβ n,k =0, 1 are the coefficients of the binary representa-tion of the indexk such that

k = β0,k+ 2β1,k+· · ·+ 2L −2β L −2,k (26) The functionS(t) is given by

S(t) =

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

sin

2πhq(t)

sin

πh −2πhq(t − LT)

(27)

5.4 Receiver design

In [27], it is shown that an optimal CPM receiver can be

built based on the Laurent linear representation and a Viterbi

detector Without going into details, we mention that

suf-ficient statistics for the decision can be obtained by

sam-pling at timesnT the outputs of 2 L −1matched filtersC k( − t);

k =0, 1, , 2 L −1−1 simultaneously fed by the complex

in-putr(t).

As we aim at bit rates higher than 1 Gbps using

low-power receivers, the complexity of this type of receivers is not

acceptable Fortunately, the Laurent approximation allows

us to construct linear near-optimum MSK-type receivers In

(24), the pulse described by the component functionC0(t)

is the most important among all other componentsC k( t) Its

duration is the longest (2T more than any other component),

and it conveys the most significant part of the energy of the

signal Kaleh [27] mentions the case of GMSK withL = 4,

where more than 99% of the energy is contained in the main

pulseC0(t) It is therefore a reasonable attempt to represent

CPM using not all components, or even only one

compo-nent We study a linear receiver taking into account only the

first Laurent pulseC0(t) According to (24), the sent signal

s(t) can thus be written as

s(t) =

e jπhα0 [n] C0(t − nT) + (t), (28)

where(t) is a negligible term generated by the pulses C k( t);

k =1, , 2 L −1−1 The received signalr(t) can be written as

where h(t) is the linear multipath channel and n(t) is the

complex-valued AWGN The equalization of the multipath

channel is done with a simple zero-forcing filter fZF(t)

as-suming perfect channel knowledge The output of the ZF

fil-ter can thus be written as



Substituting (28) in (30), we get



s(t) =

e jπhα0 [n] C0(t − nT) + (t) + n(t)  fZF(t).

(31) The output y(t) of the filter matched to C0(t) can now be

written as

y(t) =

∞

−∞s(s) · C0(s − t)ds, (32) and this signal sampled att = nT becomes

y[n]def= y(t = nT) =

∞

−∞s(s) · C0(s − nT)ds. (33) Substituting (31) in (33), we get

y[n] =

e jπhα0 [m]

∞

−∞ C0(s − mT) · C0(s − nT)ds + ξ[n],

(34)

Table 5: System parameters OQPSK

Filter bandwidth BW=1 GHz Sample period T =1 ns Number of bits per symbol 2 Number of symbols per block 256 Cyclic prefix length 64 Roll-off transmit filter 0.2

r(t)

fZF (t) s(t)

C0 (− t) y(t) y[n]

nT

Threshold detector

e jπh α0 [n]

Decoder



d[n]

Figure 9: Linear GMSK receiver using the Laurent approximation

where

ξ[n] =

∞

−∞

(s) + n(s)  fZF(s) · C0(s − nT)ds. (35)

The linear receiver presented in [27] includes a Wiener estimator, asC0(t) extends beyond t = T and thus causes

intersymbol interference (ISI) When h = 0.5 though,

e jπhα0 [m] = j α0 [m]is alternatively purely real and purely imag-inary, so the ISI in adjacent intervals is orthogonal to the sig-nal in that interval As the power in the autocorrelation of

C0(t) at t1− t2 ≥2T is very small, we can further simplify

our receiver by neglecting the ISI Equation (34) is indeed approximately:

y[n] ≈ e jπhα0 [n]+ξ [n]. (36)

Thus we get an estimate of the complex coefficient ejπh α 0 [n]

of the first Laurent pulseC0(t) after the threshold detector.

Taking into account the precoder (22), the Viterbi detection can now be replaced by a simple decoder [24]



d[n] = j − n · e jπhα0 [n] (37)

This linear receiver is shown inFigure 9

6 NUMERICAL RESULTS

6.1 Simulation setup

6.1.1 Offset-QPSK with FDE

The system parameters of OQPSK are summarized in Table 5 The root-raised cosine transmit filter has a band-width of 1 GHz The sample period after the insertion of the

CP is 1 nanosecond An OQPSK symbol carries the infor-mation of 2 bits The CP length has been set to 64 samples, which is larger than the maximum channel time dispersion (around 40 nanoseconds) The transmitter filter has a

roll-off factor of 0.2 This configuration enables a bit rate equal to 1.6 Gbps

10 8

6 4

2 0

ReceivedE b /N0 (dB)

10−4

10−3

10−2

10−1

10 0

AWGN, Roll-o ff Tx=0.2

AWGN bound

FS no eq.

NFS no eq.

(a)

25 20

15 10

5 0

Average receivedE b /N0 (dB)

10−4

10−3

10−2

10−1

10 0

Indoor multipath channel at 60 GHz, Roll-o ff Tx=0.2

AWGN bound Rayleigh bound FS-MMSE

NFS-MMSE FS-ZF NFS-ZF (b)

Figure 10: Uncoded BER performance of OQPSK with FDE for different receivers

10 8

6 4

2 0

ReceivedE b /N0 (dB)

10−4

10−3

10−2

10−1

10 0

AWGN with and without precoder

AWGN bound

Viterbi with precoder

Linear with precoder

Viterbi without precoder Linear without precoder (a)

25 20

15 10

5 0

Average receivedE b /N0 (dB)

10−4

10−3

10−2

10−1

10 0

Indoor multipath @60 GHz-ZF equalizer

AWGN bound Viterbi ZF receiver Linear ZF receiver

(b) Figure 11: BER performance of CPM with ZF equalizer for different receivers